Gauging the Spacetime Metric – Looking Back and Forth a Century Later Version 01/11/2019
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Gauging the spacetime metric – looking back and forth a century later version 01/11/2019 Erhard Scholz Abstract H. Weyl’s proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called Weyl geometry) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this research program and after the rise of a convincing alternative for the gauge idea by translating it to the phase of wave functions and spinor fields in quantum mechanics, Weyl no longer considered the original scale gauge as physically relevant. About the middle of the last century the question of conformal and/or local scale gauge transformation were reconsidered by different authors in high energy physics (Bopp, Wess, et al.) and, independently, in gravitation theory (Jordan, Fierz, Brans, Dicke). In this context Weyl geometry attracted new interest among different groups of physicists (Omote/Utiyama/Kugo, Dirac/Canuto/Maeder, Ehlers/Pirani/Schild and others), often by hypothesizing a new scalar field linked to gravity and/or high energy physics. Although not crowned by immediate success, this “retake” of Weyl geometrical methods lives on and has been extended a century after Weyl’s first proposal of his basic geometrical structure. It finds new interest in present day studies of elementary particle physics, cosmology, and philosophy of physics. arXiv:1911.01696v1 [physics.hist-ph] 5 Nov 2019 University of Wuppertal, Faculty of Math./Natural Sciences, and Interdisciplinary Centre for History and Philosophy of Science, e-mail: [email protected] 1 Contents Gauging the spacetime metric – looking back and forth a century later version 01/11/2019 .......................................... 1 Erhard Scholz 1 Introduction...................................... ........ 4 2 Weyl’sscalegaugegeometry1918–1930................. ..... 6 2.1 Purelyinfinitesimalgeometry,1918-1923 ............ 6 2.2 WithdrawelofscalegaugebyWeylafter1927/29....... 13 3 AnewstartforWeylgeometricgravityinthe1970s ........ .... 16 3.1 Newinterestsin localscaleand conformaltransformations 16 3.2 Weyl geometric gravity with a scalar field and dynamical scale connection: Omote, Utiyama, Dirac . 20 3.3 Foundationsofgeneralrelativity:EPS .............. .. 24 3.4 Geometrizingquantummechanicalconfigurationspaces 27 4 Interlude......................................... ........ 29 4.1 Basicconcepts,notationandIWG.................... 29 4.2 Weylgeometricgravity,inparticularinIWG .......... 32 5 Interesttoday..................................... ........ 35 5.1 Philosophicalreflectionsongravity ................ .. 35 5.2 Matveev/TrautmanonEPS.......................... 37 5.3 Howdoesthestandardmodelrelatetogravity?......... 39 5.4 Openquestionsincosmologyanddarkmatter.......... 43 6 Anextremelyshortglancebackandforth ................ ..... 48 References......................................... ............ 49 3 1 Introduction In 1918 Hermann Weyl proposeda generalizationof Riemannian geometry,which he consideredas betteradaptedto the field theoreticcontextof generalrelativity than the latter. His declared intention was to base geometry on purely “local” concepts which at the outset would not allow to compare field quantities X p and X p at distant ( ) ( ′) points p and p′ of the spacetime manifold, but only for infinitesimally close ones. The possibility of comparing directly two quantities at distant points appearedto him a remnant of Euclidean geometry, which Riemannian geometry had inherited via the Gaussian theory of surfaces. In Weyl’s view Riemann had generalized the latter without putting the comparability of quantities at different locations into question. He demanded in contrast that . only segments at the same place can be measured against each other. The gauging of segments is to be carried out at each single place of the world (Weltstelle), this task cannot be delegated to a central office of standards (zentrales Eichamt). (Weyl, 1919, p. 56f., emph. ES)1 Hethereforeconsideredageometryformalizedbyaconformal (pseudo-Riemannian) structureas more fundamentalthan Riemannian geometryitself (Weyl, 1918c,p.13). But it has to be supplemented by a principle which allows for comparing metrical quantities at infinitesimally close points (p , p′), realized by a principle of metrical transfer. A conformal manifold could be qualified as “metrically connected” only if a comparison of metrical quantities at different points is possible: A metrical connection from point to point is only being introduced into it [the manifold, ES], if a principle of transfer for the unit of length from one point P to an infinitesimally close one is given. (Weyl, 1918c, 14)2 Weyl formulated this principle of transfer by introducing what today would be called a connection in the local scaling group R+, i.e., by a real differential 1-form. The new type of “purely infinitesimal geometry” (Weyl’s terminology), later called Weyl geometry, was builtupon thetwo interrelated basic conceptsofa conformal structure and a scale connection as the principle of transfer. Weyl called the latter a length connection. Both were united by the possibility of changing the metrical scale by gauge transformations in the literal sense (see sec. 2.1). For a few years Weyl tried to build a unified field theory of electromagnetism and gravity upon such a geometrical structure, and to extend it to a field theory of matter.3 But in the second half of the 1920s he accepted and even contributed actively to reformulating the gauge idea in the context of the rising new quantum mechanics. Several decades later this idea was generalized to non-abelian groupsand became a fundamental conceptual ingredient even for the lager development of high energy 1 “. nur Strecken, die sich an der gleichen Stelle befinden, lassen sich aneinander messen. An jeder einzelnen Weltstelle muß die Streckeneichung vorgenommen werden, diese Aufgabe kann nicht einem zentralen Eichamt übertragen werden” (Weyl, 1919, p. 56f.). (Translation here and in the following by ES, if no reference to a published English translation is given.) 2 Emphasis in the original, here and in other places, where not explicitly stated that it is by ES. 3 Cf. (Vizgin, 1994; Goenner, 2004). 4 physics.4 In the meantime (during the 1940s) Weyl had recanted the importance of scaling transformations (localized “similarities” as he used to call them) for the search of what he called the “physical automorphisms of the world” (see sec. 2.2). The basic idea underlying Weyl’s “purely infinitesimal geometry” of 1918 reap- peared independently at several occasions during the second half of the 20th century. In the early 1960s Carl Brans and Robert Dicke developed their program of a modi- fied general relativistic theory of gravity with a non-minimally coupled scalar field. Dicke stated as an “evident” principle (which it was not, at least not for everyone): It is evident that the particular values of the units of mass, length, and time employed are arbitrary and that the laws of physics must be invariant under a general coordinate dependent change of units. (Dicke, 1962, p. 2163, emph. ES) This was very close to Weyl’s view in 1918, but Dicke postulated local scale in- variance without the complementary structure of a scale connection. As a result Brans, Dicke’s PhD student, and Dicke himself developed a theory of gravity which had an implicit relationship to the special type of Weyl geometry with an integrable scale connection, in short integrable Weyl geometry (IWG). Other authors made this relationship explicit and generalized it to the non-integrable case. This was part of a classical field theory program of gravity research, but the importance of conformal transformations got also new input from high energy physics. And even the original form of Weyl geometry had some kind of re- vival from the 1960s onward in scalar field theories of gravity or nuclear struc- tures, initiated by authors in Japan (Omote/Utiyama/Kugo) and, independently Eu- rope/USA (Dirac/Canuto/Maeder), and also in the foundational studies of gravity (Ehlers/Pirani/Schild). This restart in the last third of the 20th century of research building on, and extending, Weyl geometric methods in physics has lasted until today and shows that Weyl’s disassociation from his scale gauge idea is not at all to be considered a final verdict on his geometrical ideas developed between 1918 and the early 1920s. The following paper tries to give an account of the century long development from Weyl’s original scale gauge geometry of 1918 (and the reasons why he thought it an important improvement on the earlier field theories building upon Riemannian geometry), through its temporary disregard induced by the migration of the gauge idea frommetrical scale to quantumphase (ca. 1930–1960,section 2), and the revival in the early 1970s indicated above (section 3), to a report on selected research in physics, which uses Weyl geometric methods in a crucial way (section 5). Basic concepts and notations of Weyl geometric gravity (in the moderately modernized form in which they are used in section 5) are explained in an interlude between the historical account and the survey of present studies (sec. 4). Short reflections on this glance back and forth are given at the end of the paper (section 6). 4 See N. Straumann’s contribution to this volume. 5 2 Weyl’s scale gauge geometry 1918–1930 2.1 Purely infinitesimal geometry, 1918-1923 Parallel to finalizing